{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = np.loadtxt('./ex1data1.txt',delimiter=',', dtype='float32')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "data_t = np.append(np.reshape(np.ones(len(data)),[len(data),1]), data, axis=1)\n",
    "X = data_t[...,0:2]\n",
    "data_x = data_t[...,1:2]\n",
    "data_y = data_t[...,2:3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1a9afef3dc8>]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
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     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.xlabel('Population of city in 10,000s')\n",
    "plt.ylabel('Profit in $10,000s')\n",
    "plt.grid(True)\n",
    "plt.plot(data_x, data_y, 'r.')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def costValue(x_i, y_i, theta): #计算损失函数的值\n",
    "    x = np.mat(x_i)\n",
    "    y = np.mat(y_i)\n",
    "    cost = np.power(((x * theta)- y),2)\n",
    "    return np.sum(cost)/(2*len(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_theta(data, a, iterator)->np.ndarray: #梯度下降求theta\n",
    "    temp = np.zeros((data.shape[1]-1,1)) #创立theta 为0的向量. np.zeros((行，列))\n",
    "    theta = temp\n",
    "    cost = np.zeros(iterator)\n",
    "    #temp theta (2 * 1)\n",
    "    x_i = data[...,0:2] # (97 * 2)\n",
    "    y_i = data[...,2:3] # (97 * 1)\n",
    "    # 假设函数 h(x) = theta_0 + theta_1 * x\n",
    "    #则 J(theta)计算如下\n",
    "  \n",
    "    m = x_i.shape[0]\n",
    "    \n",
    "    for i in range(iterator):\n",
    "        for j in range(0, x_i.shape[1]):\n",
    "            error = 0\n",
    "            for k in range(m):\n",
    "                error += (theta[0] * data[k,0] + theta[1] * data[k, 1] - data[k, 2])*data[k,j]\n",
    "            temp[j] = theta[j] - (a / m) * error\n",
    "        theta = temp\n",
    "        \n",
    "        cost[i] = costValue(x_i, y_i, theta)\n",
    "                            \n",
    "    return theta,cost"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_theta_2(X,y, a, iterator) -> np.ndarray:  # 梯度下降求theta\n",
    "    X = np.mat(X)\n",
    "    temp = np.matrix(np.zeros(X.shape[1]))  # 初始化一个 θ 临时矩阵(1, 2)\n",
    "    theta = temp\n",
    "    cost = np.zeros(iterator)  # 初始化一个ndarray，包含每次epoch的cost\n",
    "    m = X.shape[0]  # 样本数量m\n",
    "\n",
    "    for i in range(iterator):\n",
    "        # 利用向量化一步求解\n",
    "        temp = theta - (a / m) * (X * theta.T - y).T * X\n",
    "\n",
    "        theta = temp\n",
    "        cost[i] = costValue(X, y, theta.T)\n",
    "    return theta,cost"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(97, 2)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.shape"
   ]
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  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "def NormalEquation(X,y): #正规方程求解theta\n",
    "\n",
    "    X = np.matrix(X)\n",
    "    y = np.matrix(y)\n",
    "    theta = (X.T * X).I * X.T * y\n",
    "    return theta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "theta, cost= get_theta(data_t, 0.01, 1000)\n",
    "theta_2, cost_2 = get_theta_2(X,data_y, 0.01, 1000)\n",
    "theta_3 = NormalEquation(X,data_y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([[-3.25088245],\n",
       "        [ 1.12836318]]),\n",
       " matrix([[-3.24140238,  1.12729424]]),\n",
       " matrix([[-3.89578118],\n",
       "         [ 1.19303369]]))"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta,theta_2,theta_3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "yy = data_t[...,0:2].dot(theta)\n",
    "yy_2 = data_t[...,0:2].dot(theta_2.T)\n",
    "yy_3 = data_t[...,0:2].dot(theta_3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x1a9b0841f08>"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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       "     <g transform=\"translate(77.603125 53.633502)scale(0.1 -0.1)\">\r\n",
       "      <use xlink:href=\"#DejaVuSans-116\"/>\r\n",
       "      <use x=\"39.208984\" xlink:href=\"#DejaVuSans-104\"/>\r\n",
       "      <use x=\"102.587891\" xlink:href=\"#DejaVuSans-101\"/>\r\n",
       "      <use x=\"164.111328\" xlink:href=\"#DejaVuSans-116\"/>\r\n",
       "      <use x=\"203.320312\" xlink:href=\"#DejaVuSans-97\"/>\r\n",
       "      <use x=\"264.599609\" xlink:href=\"#DejaVuSans-95\"/>\r\n",
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       "     </g>\r\n",
       "    </g>\r\n",
       "   </g>\r\n",
       "  </g>\r\n",
       " </g>\r\n",
       " <defs>\r\n",
       "  <clipPath id=\"pe04bbe8ea9\">\r\n",
       "   <rect height=\"217.44\" width=\"334.8\" x=\"40.603125\" y=\"7.400689\"/>\r\n",
       "  </clipPath>\r\n",
       " </defs>\r\n",
       "</svg>\r\n"
      ],
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.xlabel('Population of city in 10,000s')\n",
    "plt.ylabel('Profit in $10,000s')\n",
    "plt.grid(True)\n",
    "plt.plot(data_x, data_y, 'r.')\n",
    "plt.plot(data_x, yy, 'g',label='theta')\n",
    "plt.plot(data_x,yy_2,'b',label='theta_2')\n",
    "plt.plot(data_x,yy_3, 'y', label='theta_3')\n",
    "plt.legend(loc=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 多变量拟合"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "def featureScaling(data):\n",
    "    for col in data.columns:\n",
    "        data[col] = (data[col]-data[col].min()) / (data[col].max() - data[col].min())\n",
    "    return data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "data_m = pd.read_table('./ex1data2.txt', sep=',',header=None,names=['size','num','price'])\n",
    "data_m_f = featureScaling(data_m)\n",
    "data_m.insert(0,'bias',np.ones(len(data_m)))\n",
    "data_m_array = data_m_f.values\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_m = data_m_array[...,0:data_m_array.shape[1]-1]\n",
    "y_m = data_m_array[...,data_m_array.shape[1]-1:data_m_array.shape[1]]\n",
    "theta_m_1,cost = get_theta_2(x_m,y_m, 0.1, 1000)\n",
    "theta_m_2 = NormalEquation(x_m,y_m)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(matrix([[0.029095  , 0.88642184, 0.02111417]]),\n",
       " matrix([[ 0.05578752],\n",
       "         [ 0.95241114],\n",
       "         [-0.06594731]]))"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta_m_1,theta_m_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1a9b08c31c8>]"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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